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Ramanujan surprises again

By

Marianne Freiberger

Submitted by Marianne on November 3, 2015

Ramanujan's manuscript. The representations of 1729 as the sum of two cubes appear in the bottom right corner. The equation expressing the near counter examples to Fermat's last theorem appears further up: α3 + β3 = γ3 + (-1)n. Image courtesy Trinity College library. Click here to see a larger image.

A box of manuscripts and three notebooks. That's all that's left of
the work of Srinivasa Ramanujan, an Indian mathematician who lived
his remarkable but short life around the beginning of the
twentieth century. Yet, that small stash of mathematical legacy still
yields surprises. Two mathematicians of Emory University, Ken
Ono and Sarah Trebat-Leder,
have recently made a fascinating discovery within its yellowed pages. It shows
that Ramanujan was further ahead of his time than anyone had
expected, and provides a beautiful link between several milestones in the history of
mathematics. And it all goes back to the innocuous-looking number 1729.

Ramanujan's story is as inspiring as it is tragic. Born in 1887 in a small village around 400 km from
Madras (now Chennai), Ramanujan developed a passion for mathematics at a young age, but had
to pursue it mostly alone and in poverty. Until, in 1913, he decided to write a
letter to the famous Cambridge number theorist G.H. Hardy. Accustomed
to this early form of spam, Hardy might have been
forgiven for dispatching the highly
unorthodox letter straight to the bin. But he
didn't. Recognising the author's genius, Hardy invited Ramanujan to Cambridge,
where he arrived in 1914. Over the following years, Ramanujan more
than repaid Hardy's faith in his talent, but suffered ill health due, in part, to the
grizzly English climate and food. Ramanujan returned to India in
1919, still feeble, and died
the following year, aged only 32. Hardy later described his collaboration with
Ramanujan as "the one romantic incident in my life".

The taxi-cab number

The romanticism rubbed off on the number 1729, which plays a
central role in the Hardy-Ramanujan story. "I remember once going to see [Ramanujan]
when he was ill at Putney," Hardy wrote later. "I had ridden in taxi
cab number 1729 and remarked that the number seemed to me rather a
dull one, and that I hoped it was not an unfavourable omen. 'No', he
replied, 'it is a very interesting number; it is the smallest number
expressible as the sum of two cubes in two different ways.'" What
Ramanujan meant is that

The anecdote gained the number 1729 fame in mathematical circles, but until
recently people believed its curious property was just another random
fact Ramanujan carried about in his brain — much like a train spotter
remembers train arrival times. What Ono and Trebat-Leder's discovery
shows, however, is that it was just the tip of an ice berg. In reality
Ramanujan had been busy developing a theory that was several decades ahead
of its time and yields results that are interesting to mathematicians even today. He just didn't live long enough to publish
it.

The discovery came when Ono and fellow mathematician Andrew Granville were
leafing through Ramanujan's manuscripts, kept at the Wren Library at
Trinity College, Cambridge. "We were sitting right next to the
librarian's desk, flipping page by page
through the Ramanujan box," recalls Ono. "We came across this one
page which had on it the two representations of 1729 [as the sum of
cubes]. We started laughing immediately."

Fermat's last theorem and near misses

Srinivasa Ramanujan (1887 - 1920).

Ono and Granville spotted the famous number even though it didn't
itself appear on the page. Ramanujan had only written down the equation But what delighted the two mathematicians more than meeting
the famous number in disguise was another equation appearing on the same
page. It clearly showed that Ramanujan had been working on a
problem that had become notorious way back in the 17th century and whose solution, in
the 1990s, was a major mathematical sensation. It's known as
Fermat's last theorem.

The problem, like so many problems in number theory, is easy to understand. Pythagoras’ theorem tells us that if a right-angled triangle has sides of lengths and with being the longest side, then the three lengths satisfy the equation

There are infinitely triples of positive whole numbers and which satisfy this relationship. An example is and A natural question is whether you can also find three positive whole numbers (excluding 0s) satisfying the equation

or

or

and so on.

In 1637 the French mathematician Pierre
de Fermat confidently asserted that the answer is no. If is a whole number greater than then there are no positive whole number triples and such that

Fermat scribbled in the margin of a page in a book that he had "discovered a truly marvellous proof of this, which this margin is too narrow to contain". Naturally, this assertion was like catnip to mathematicians, who subsequently drove themselves crazy, for over 350 years, trying to find this "truly marvellous proof".

What the equation in Ramanujan’s manuscript illustrates is that Ramanujan had found a whole family (in fact an infinite family) of positive whole number triples and that very nearly, but not quite, satisfy Fermat’s famous equation for They are off only by plus or minus one, that is, either

or

Since any positive whole number triple satisfying the equation would render Fermat’s assertion (that there are no such triples) false, Ramanujan had pinned down an infinite family of near-misses of what would be counter-examples to Fermat’s last theorem.

"None of us had any idea that Ramanujan was thinking about anything
[remotely] related to Fermat's last theorem," says Ono. "But here on a page, staring
us in the face, were infinitely many near counter-examples to it, two
of which happen
to be related to 1729. We were floored." Even today,
nearly 400 years after Fermat's claim and 20 years after its
resolution, only a handful of mathematicians even know about the
family Ramanujan had come up with. "I'm a Ramanujan scholar and I wasn't aware of
it," says Ono. "Basically, nobody knew."

Elliptic curves and climbing K3.

But this isn't all. When Ono and his graduate student Sarah
Trebat-Leder decided to investigate further, looking at other pages in
Ramanujan's work, they found he had developed a sophisticated
mathematical theory that went beyond anything people had suspected. "Sarah and I spent time thinking more deeply about what Ramanujan had
really done, and it turns out that he anticipated [an area of] mathematic 30 or 40
years before anyone knew this field would exist. That's what we are
excited about."

It turns out that from looking at equations of the form

it’s not too large a mathematical step to considering equations of the form

where , and are constants. If you plot the points that satisfy such an equation (for given values of and ) in a coordinate system, you get a shape called an elliptic curve (the precise definition is slightly more
involved, see here). Elliptic
curves played an important role in the eventual proof of Fermat's last theorem,
which was delivered in the 1990s by the mathematician Andrew
Wiles.

Ono and Trebat-Leder found that Ramanujan had also delved into the
theory of elliptic curves. He did not anticipate the path taken by
Wiles, but instead discovered an object that is more
complicated than elliptic curves. When objects of this kind were rediscovered around
forty years later they were adorned with the name of
K3 surfaces — in honour of the mathematicians Ernst
Kummer, Erich
Kähler and Kunihiko
Kodaira, and the mountain K2, which is as difficult to climb as K3
surfaces are difficult to handle mathematically.

That Ramanujan should have discovered and understood an exceedingly complicated K3 surface is in itself remarkable. But his work on the surface also provided an unexpected gift to Ono and Trebat-Leder, which links back to elliptic curves. Like all equations, any elliptic curve equation

naturally cries out for solutions: pairs of numbers that satisfy the equation. In the spirit of Fermat, you might look for whole number solutions, but number theorists usually give themselves a little more leeway. They look for solutions that are rational numbers, that is, numbers that can be written as fractions.

The elliptic curves corresponding to whole number values of a between -2 and 1 and whole number values of values of b between -1 and 2. Only the curve for a = b = 0 doesn't qualify as an elliptic curve because it has a sharp corner.

Last year, in 2014, the
mathematician Manjul Bhargava won the Fields Medal, one of the highest
honours in mathematics, for major progress in this context. Bhargava
showed that most elliptic curves fall into one of two particularly simple classes. Either there are only finitely many rational number solutions; or there are infinitely many, but there is a recipe that produces all of them from just a single rational number solution.
(You can read our interview with Bhargava and our article exploring some of his work.)

If you sift through all elliptic curves in a systematic way, for example by ordering them according to the size of the constants and that appear in their formulas, then you are most likely only ever going to come across these "simple" elliptic curves. The probability of finding a more complicated one, which requires two or three solutions to generate them all, is zero. Searching for such elliptic curves systematically is like searching a haystack for a needle in a way that guarantees the needle will always slip through the net. To get at those more complicated elliptic curves, you need another method.

And this is exactly what Ramanujan came up with. His work on the K3 surface he
discovered provided Ono and Trebat-Leder
with a method to produce, not just one, but infinitely many elliptic
curves requiring two or three solutions to generate all other
solutions. It's not the first method that has been found, but it required no effort. "We tied the world record on the problem [of finding such
elliptic curves], but we didn't
have to do any heavy lifting," says Ono. "We
did next to nothing, expcet recognise what Ramanujan did."

Physics and extra dimensions

There is another interesting twist to this story. While Ramanujan
was working in the abstract realms of number theory, physicists
studying real-world phenomena began developing the theory of quantum
mechanics. Although a triumph in its own right, it soon became clear
that the resulting quantum physics clashed with existing physical theories in an
unredeemable way. The rift still hasn't been healed and
presents the biggest problem of twenty-first century physics (see here
to find out more). One attempt at rescuing the situation was the
development, started in the 1960s, of string theory, a prime
candidate for a "theory of everything" uniting the disparate strands
of modern physics.

G.H. Hardy (1877 - 1947).

A curious prediction of string theory is that the world we live in
consists of more than the three spatial dimensions we can see. The
extra dimensions, the ones we can't see, are rolled up
tightly in tiny little spaces too small for us to perceive. The theory
dictates that those tiny little spaces have a particular geometric
structure. There's a class of geometric objects, called
Calabi-Yau manifolds, which fits the bill (see this article to find out more). And
one of the
simplest classes of Calabi-Yau manifolds comes from, wait for it,
K3 surfaces, which Ramanujan was the first to discover.

Ramanujan could never have dreamt of this development, of course. "He was a whiz with formulas and I think
[his aim was] to construct those near counter-examples to Fermat's
last theorem." says Ono. "So he
developed a theory to find these
near misses, without recognising that the
machine he was building, those formulas that he was writing down,
would be useful for anyone, ever, in the future."

Ono doesn't rule out that Ramanujan's manuscripts contain further
hidden treasures. "I've known about 1729 for thirty years. It's a lovely, romantic
number. Ramanujan was a genius and we are still
learning about the extent to which his creativity led him to his
formulas. His work amounts to one box, kept at Trinity College, and
three notebooks, kept at the University of
Madras. That's not a lot. It's crazy that we are still figuring out
what he had in mind. When is it going to end?"

Of course, the Hardy & Ramanujam story led to a series of pairs of cubes being combined in n ways (n=2 for 1729) being termed "Taxicab numbers". See a paper my father, the late Edwin Rosenstiel, which announed the discovery of the fourth in the series, published in The Institute of Mathematics and its Applications Bulletin July, 1991, Volume 27, pp 155-157. The other authors were John Dardis and myself (I did the computing on PCs of the era). My parents lived in Putney. The Hardy & Ramanujam story inspired the search that led to the paper.

The smallest number that can be written as a sum of two cubes of integers in two (non-trivially) different ways is 91, not 1729. 91 = 3^3 + 4^3 = 6^3 + (-5)^3. I wonder if Ramanujan specified to Hardy that he was specifically referring to the natural numbers, because the folklore never seems to specify that.

Sir,
Going by your logic , there would not be any 'smallest number' .
E.g. -91 = (-3) ^3 + (-4) ^3 = (-6) ^3 + (5) ^3 < 91
and so the negative of any larger such positive numbers will give further smaller numbers (e.g. -1729) and so on ............
So I think this condition you are saying has to be implicitly implied :)

This may have been pointed out elsewhere, but Ramanujan's statement was not that 1729 was the smallest number that satisfied the sum of two integer cubes; he maintained it was the smallest which could thus be described by two different pairs of integers. Therefore the 91 example does not apply.

Agree with Raul's statement above. This is a great discovery made by Ramanujan. A cube can be found by the sum of three cubes where one of the cubes is always the cube of 1!! Amazing!!! I think it may be that this can be extended to further developments including a cube being the sum of four cubes or more with perhaps a constant cube in place, perhaps other than cube of 1... Kaiser Tarafdar(Math enthusiast)

That is a tremendous finding regarding ramanujan's discovery of a cube being the sum of three other cubes where one commonality of the cube of 1. That is a new discovery and not just a "near-miss" of Fermat's last theorem! I think ramanujan was on his way to find further extension of that with perhaps a cube being the sum of four or five or even n number of cubes with a constant or semi-constant cube in place just like the cube of 1... A genius of the ages!! Kaiser Tarafdar

There are problems that are easy to solve in theory, but impossible to solve in practice. Intrigued? Then join us on a journey through the world of complexity, all the way to the famous P versus NP conjecture.